this allows us to write
P
1 + it = (1 + rt ) -ɪ . (19)
Pt
Bringing to mind the situation mentioned above, where the monetary base growth rate is
constant, and substituting the inverse of (19) in (16), results in
P1P
bt+1 — —--- = b + gt - txt (20)
t+1 Pt 1 + rtPt+ι t
bt+1 = (1 + rt )bt + (1 + rt )(gt - txt ). (21)
Assuming also the hypothesis of a constant real interest rate, and for simplicity, if the
budget deficit is stable, (gt - txt ) = (g - tx), we have also
bt+1 = (1 + r)bt + (1 + r )(g - tx). (22)
From the the last expression, it becomes clear that bt = Bt/Pt will follow an explosive
trajectory since (1+r)>0. Notice also that from equation (22), the growth rate of
government debt is given by the following difference equation
bs
ɪ = (1 + r )(1 - _ ), (23)
bt bt
where the primary budget surplus, _, is given by _ = tx-g, and that eventually (23)
converges to (1+r) while b increases. However, in this case, the government is
conducting Ponzi games, and it would no be possible to satisfy a transversality
condition such as the one given by equation (7).
An explosive situation for the stock of real government debt will be avoided if the initial
value for b is
b0 =-(1 + r)(g - tx)/ r , (24)
in order to ensure that b remains constant at that same value. As a matter of fact, with
that initial value for b one gets simply
10
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